# purely inseparable

Let $F$ be a field of characteristic $p>0$ and let $\alpha$ be an element which is algebraic over $F$. Then $\alpha$ is said to be purely inseparable over $F$ if $\alpha^{p^{n}}\in F$ for some $n\geq 0$.

An algebraic field extension $K/F$ is purely inseparable if each element of $K$ is purely inseparable over $F$.

Purely inseparable extensions have the following property: if $K/F$ is purely inseparable, and $A$ is an algebraic closure of $F$ which contains $K$, then any homomorphism $K\to A$ which fixes $F$ necessarily fixes $K$.

Let $K/F$ be an arbitrary algebraic extension. Then there is an intermediate field $E$ such that $K/E$ is purely inseparable, and $E/F$ is separable.

###### Example.

Let $s$ be an indeterminate, and let $K=\mathbb{F}_{3}(s)$ where $\mathbb{F}_{3}$ is the finite field with $3$ elements. Let $F=\mathbb{F}_{3}(s^{6})$. Then $K/F$ is neither separable, nor purely inseparable. Let $E=\mathbb{F}_{3}(s^{3})$. Then $E/F$ is separable, and $K/E$ is purely inseparable.

Title purely inseparable PurelyInseparable 2013-03-22 14:49:08 2013-03-22 14:49:08 mclase (549) mclase (549) 6 mclase (549) Definition msc 12F15